"Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations." - Steven Strogatz
Differential Equation Lab
A mathematical, visual and computational journey through ordinary differential equations, partial differential equations, Laplace transforms, simulations, and the mathematics of change.
A falling ball, a vibrating string, heat spreading through metal, a quantum particle evolving in time — all of these are described by differential equations.
This site approaches them both analytically and computationally: solving equations by hand, interpreting their structure, and exploring their behaviour through numerical experiments and visualizations.
This site is a guide through that world — from simple first-order equations to second-order systems, Laplace transforms, partial differential equations, and the numerical methods used when exact solutions do not exist.
The goal is simple: understand how these equations behave, and how to solve them both analytically and numerically.
First-Order ODEs
Learn the fundamental techniques for solving first-order differential equations, from direction fields and separable equations to integrating factors.
Second-Order ODEs
Learn the fundamental methods for solving linear second-order differential equations, from homogeneous and inhomogeneous equations to oscillatory systems and resonance.
Laplace Transforms
A powerful method for solving initial value problems, especially when forcing terms, discontinuities, or impulses appear.
Partial Differential Equations
From waves and heat flow to electrostatics and quantum mechanics, discover the power of partial differential equations.
Numerical Methods
When analytical solutions are unavailable, numerical methods provide powerful tools for approximating and visualizing differential equations.
Non-Dimensionalization
Scaling equations to reveal the essential dimensionless parameters that govern the behavior of a physical system.
Equations as Physical Systems
Wave Equation
\( \displaystyle \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \)
Waves, vibrations and sound.Heat Equation
\( \displaystyle \frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2} \)
Heat flow, diffusion, and equilibration.Laplace Equation
\(\nabla^2 u = 0\)
Equilibrium, potentials, and steady states.Schrödinger Equation
\( \displaystyle i\hbar \frac{\partial \psi}{\partial t} \, = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} \, + V(x) \, \psi \)
Wave functions, quantum evolution, and probability.